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Magazine Chapter 6: Problem 14
\(\frac{d^{2} y}{d x^{2}}=\sqrt[3]{3 x-1} ; y=2\) and \(y^{\prime}=5\) when \(x=3\)
Short Answer
Expert verified
y = \frac{3}{5}(3x-1)^{5/3} + 3x -8.4
Step by step solution
01
Recognize the Differential Equation
The given differential equation is a second-order ordinary differential equation (ODE): \[\frac{d^{2} y}{d x^{2}}=\frac{(3 x-1)^{1/3}}{1}\]
02
Integrate the Second Derivative
Integrate the second derivative to find the first derivative. This means solving \[\frac{d y'}{d x} = (3 x - 1)^{1/3}\] with respect to x. Let: \[u = 3x - 1, \frac{du}{dx} = 3\]. Then rewrite the integral: \[\frac{d y'}{d x} = (3 x - 1)^{1/3}\] becomes \[\frac{dy'}{dx} = u^{1/3}\frac{1}{3}\frac{du}{dx}\]. Integrate \[y' = \frac{1}{3} \frac{3^{2/3}}{2/3}u^{2/3} + C_1 \].
03
Substitute Back in Terms of x
After integration, substitute back to remove u: \[ y' = \frac{(3x-1)^{2/3}}{2} + C_1 \].
04
Apply Initial Conditions to Determine Constants
Substitute the initial conditions. Given: \[y' = 5\] when \[x = 3\]. This equation solves: \[ 5 = \frac{(3 \times 3 - 1)^{2/3}}{2} + C_1 \]. So, \[ 5 = \frac{(8)^{2/3}}{2} + C_1\]. Simplifying this, \[ 5 = \frac{4}{2} + C_1 \rightarrow C_1 = 5 - 2 = 3 \].
05
Integrate Again to Find y
Integrate y' to determine y: \[ y = \frac{3}{5}(3x-1)^{5/3} + 3x + C_2 \].
06
Apply Second Initial Condition at x = 3
Given y = 2 when x = 3: \[ 2 = \frac{3}{5}(8)^{5/3} + 3(3) + C_2 \]. Simplify and solve for C_2: \[ 2 = 4.8\frac{3}{5} + 9 + C_2\rightarrow C_2 = 2 - 19.2 -9 = -8.4\] . Thus, C2 is found; the second integral gives the complete solution for y.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration is a central technique used to solve ordinary differential equations (ODEs). When dealing with a second-order ODE, like \(\frac{d^{2} y}{d x^{2}}=\frac{(3 x-1)^{1/3}}{1}\), the goal is to find a function \(y(x)\) that satisfies this equation. This often requires integrating the second derivative twice.
First, integrate \(\frac{d y'}{d x} = (3 x - 1)^{1/3}\) to find \(y'(x)\).
To simplify the integral, we can use a substitution: let \(u = 3x - 1\), so \(\frac{du}{dx} = 3\). Substituting gives us \(u^{1/3}\frac{1}{3}\frac{du}{dx}\).
This integral simplifies to \(y' = \frac{1}{3} \frac{3^{2/3}}{2/3}u^{2/3} + C_1 \), where \(C_1\) is the integration constant. Substituting back gives \(y' = \frac{(3x-1)^{2/3}}{2} + C_1 \).
The same steps are repeated to integrate \(y'(x)\) to find \(y(x)\).
First, integrate \(\frac{d y'}{d x} = (3 x - 1)^{1/3}\) to find \(y'(x)\).
To simplify the integral, we can use a substitution: let \(u = 3x - 1\), so \(\frac{du}{dx} = 3\). Substituting gives us \(u^{1/3}\frac{1}{3}\frac{du}{dx}\).
This integral simplifies to \(y' = \frac{1}{3} \frac{3^{2/3}}{2/3}u^{2/3} + C_1 \), where \(C_1\) is the integration constant. Substituting back gives \(y' = \frac{(3x-1)^{2/3}}{2} + C_1 \).
The same steps are repeated to integrate \(y'(x)\) to find \(y(x)\).
Initial Conditions
Initial conditions are given values that help determine the constants in the solution of a differential equation. For our ODE, the initial conditions are \(y=2\) and \(y'=5\) when \(x=3\). These values are used to solve for \(C_1\) and \(C_2\) once we integrate.
After the first integration, we substitute \(x=3\) and \(y'=5\) into the equation \(y' = \frac{(3x-1)^{2/3}}{2} + C_1\) to find \(C_1\). Simplifying this, we get \(C_1 = 3\).
Next, after integrating again to find \(y(x)\), we substitute the initial condition \(y=2\) and \(x=3\) into \(y = \frac{3}{5}(3x-1)^{5/3} + 3x + C_2\) to solve for \(C_2\). This gives us \(C_2 = -8.4\).
After the first integration, we substitute \(x=3\) and \(y'=5\) into the equation \(y' = \frac{(3x-1)^{2/3}}{2} + C_1\) to find \(C_1\). Simplifying this, we get \(C_1 = 3\).
Next, after integrating again to find \(y(x)\), we substitute the initial condition \(y=2\) and \(x=3\) into \(y = \frac{3}{5}(3x-1)^{5/3} + 3x + C_2\) to solve for \(C_2\). This gives us \(C_2 = -8.4\).
Ordinary Differential Equations
An ordinary differential equation (ODE) relates a function and its derivatives. A second-order ODE involves the second derivative of the function. The given equation, \(\frac{d^{2} y}{d x^{2}} = (3 x - 1)^{1/3}\), is a second-order ODE.
To solve a second-order ODE, we generally:
To solve a second-order ODE, we generally:
- Integrate the second derivative to find the first derivative.
- Apply initial conditions to determine integration constants.
- Integrate the first derivative to find the function.
- Use initial conditions again to solve for any remaining constants.
